From Pope's turbulence book, for homogeneous isotropic turbulence, how to prove the following relation for strain rate energy spectrum

$$\langle S_{ij}S_{ij} \rangle = \int_{0}^{\infty} k^2 E(k)dk $$

where $$S_{ij} = \dfrac{1}{2}(u_{i,j}+u_{j,i})$$ and $$E(k) = 2\pi k^2 \Phi_{ii}(k,t)$$ where

$$\Phi_{ij}({k},t) = \dfrac{1}{(2\pi)^3} \int_{\mathbb{R}^3} e^{-i{k}\cdot {r}} R_{ij}({r} ,t ) d{r}$$ and

$$ R_{ij}({r},{x},t) = \langle u_i({x},t) u_j({x+r},t) \rangle$$

  • $\begingroup$ Two quick questions: (1) is the notation $Q_{i,j} = \partial_{j} Q_{i}$ and (2) in the $E(k)$ formula, are the indices of $\Phi$ supposed to be the same or should it be $ij$? $\endgroup$ – honeste_vivere Feb 2 '18 at 15:29
  • $\begingroup$ As a hint, the Fourier transform of the autocorrelation is the power spectral density, by definition. $\endgroup$ – honeste_vivere Feb 2 '18 at 16:02

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